Locally Linearized Runge Kutta method of Dormand and Prince

In this paper, the effect that produces the Local Linearization of the embedded Runge-Kutta formulas of Dormand and Prince for initial value problems is studied. For this, embedded Locally Linearized Runge-Kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations. For a variety of well-known test equations with different dynamics, the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones, which implies a substantial reduction of the number of time steps and, consequently, a sensitive reduction of the overall computational cost of their adaptive implementation.

[1]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[2]  Juan C. Jiménez,et al.  Local Linearization - Runge-Kutta methods: A class of A-stable explicit integrators for dynamical systems , 2012, Math. Comput. Model..

[3]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[4]  Julyan H. E. Cartwright,et al.  THE DYNAMICS OF RUNGE–KUTTA METHODS , 1992 .

[5]  Ian Stewart,et al.  Warning — handle with care! , 1992, Nature.

[6]  Marlis Hochbruck,et al.  Exponential Rosenbrock-Type Methods , 2008, SIAM J. Numer. Anal..

[7]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[8]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[9]  C. Hayashi,et al.  Nonlinear oscillations in physical systems , 1987 .

[10]  J. Lambert.  Numerical Methods for Ordinary Differential Equations , 1991 .

[11]  J. C. Jimenez,et al.  Rate of convergence of Local Linearization schemes for random differential equations , 2009 .

[12]  Juan C. Jiménez,et al.  Local Linearization-Runge Kutta (LLRK) Methods for Solving Ordinary Differential Equations , 2006, International Conference on Computational Science.

[13]  C. Loan,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix , 1978 .

[14]  Christian Bischof,et al.  Automatic differentiation for MATLAB programs , 2003 .

[15]  Peter Deuflhard,et al.  Recent progress in extrapolation methods for ordinary differential equations , 1985 .

[16]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[17]  T. Steihaug,et al.  An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations , 1979 .

[18]  A. R. Humphries,et al.  Dynamical Systems And Numerical Analysis , 1996 .

[19]  Juan C. Jiménez,et al.  A higher order local linearization method for solving ordinary differential equations , 2007, Appl. Math. Comput..

[20]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[21]  H. Zedan,et al.  Avoiding the exactness of the Jacobian matrix in Rosenbrock formulae , 1990 .

[22]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[23]  Juan C. Jiménez,et al.  Rate of convergence of local linearization schemes for initial-value problems , 2005, Appl. Math. Comput..

[24]  Juan C. Jiménez,et al.  Dynamic properties of the local linearization method for initial-value problems , 2002, Appl. Math. Comput..

[25]  Lawrence F. Shampine,et al.  Accurate numerical derivatives in MATLAB , 2007, TOMS.

[26]  Abdus Salam,et al.  LOCAL LINEARIZATION METHODS FOR THE NUMERICAL INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS: AN OVERVIEW , 2009 .

[27]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[28]  E. Hairer,et al.  Geometric Numerical Integration , 2022, Oberwolfach Reports.

[29]  J. C. Jimenez,et al.  Convergence rate of strong Local Linearization schemes for stochastic differential equations with additive noise , 2012 .

[30]  Joseph D. Skufca,et al.  Analysis Still Matters: A Surprising Instance of Failure of Runge-Kutta-Felberg ODE Solvers , 2004, SIAM Rev..

[31]  Gene H. Golub,et al.  Matrix computations , 1983 .

[32]  N. Higham.  The Scaling and Squaring Method for the Matrix Exponential Revisited , 2005, SIAM J. Matrix Anal. Appl..